31x^2-38x+11=0

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Solution for 31x^2-38x+11=0 equation:


Simplifying
31x2 + -38x + 11 = 0

Reorder the terms:
11 + -38x + 31x2 = 0

Solving
11 + -38x + 31x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
31 the coefficient of the squared term: 

Divide each side by '31'.
0.3548387097 + -1.225806452x + x2 = 0

Move the constant term to the right:

Add '-0.3548387097' to each side of the equation.
0.3548387097 + -1.225806452x + -0.3548387097 + x2 = 0 + -0.3548387097

Reorder the terms:
0.3548387097 + -0.3548387097 + -1.225806452x + x2 = 0 + -0.3548387097

Combine like terms: 0.3548387097 + -0.3548387097 = 0.0000000000
0.0000000000 + -1.225806452x + x2 = 0 + -0.3548387097
-1.225806452x + x2 = 0 + -0.3548387097

Combine like terms: 0 + -0.3548387097 = -0.3548387097
-1.225806452x + x2 = -0.3548387097

The x term is -1.225806452x.  Take half its coefficient (-0.612903226).
Square it (0.3756503644) and add it to both sides.

Add '0.3756503644' to each side of the equation.
-1.225806452x + 0.3756503644 + x2 = -0.3548387097 + 0.3756503644

Reorder the terms:
0.3756503644 + -1.225806452x + x2 = -0.3548387097 + 0.3756503644

Combine like terms: -0.3548387097 + 0.3756503644 = 0.0208116547
0.3756503644 + -1.225806452x + x2 = 0.0208116547

Factor a perfect square on the left side:
(x + -0.612903226)(x + -0.612903226) = 0.0208116547

Calculate the square root of the right side: 0.144262451

Break this problem into two subproblems by setting 
(x + -0.612903226) equal to 0.144262451 and -0.144262451.

Subproblem 1

x + -0.612903226 = 0.144262451 Simplifying x + -0.612903226 = 0.144262451 Reorder the terms: -0.612903226 + x = 0.144262451 Solving -0.612903226 + x = 0.144262451 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.612903226' to each side of the equation. -0.612903226 + 0.612903226 + x = 0.144262451 + 0.612903226 Combine like terms: -0.612903226 + 0.612903226 = 0.000000000 0.000000000 + x = 0.144262451 + 0.612903226 x = 0.144262451 + 0.612903226 Combine like terms: 0.144262451 + 0.612903226 = 0.757165677 x = 0.757165677 Simplifying x = 0.757165677

Subproblem 2

x + -0.612903226 = -0.144262451 Simplifying x + -0.612903226 = -0.144262451 Reorder the terms: -0.612903226 + x = -0.144262451 Solving -0.612903226 + x = -0.144262451 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.612903226' to each side of the equation. -0.612903226 + 0.612903226 + x = -0.144262451 + 0.612903226 Combine like terms: -0.612903226 + 0.612903226 = 0.000000000 0.000000000 + x = -0.144262451 + 0.612903226 x = -0.144262451 + 0.612903226 Combine like terms: -0.144262451 + 0.612903226 = 0.468640775 x = 0.468640775 Simplifying x = 0.468640775

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.757165677, 0.468640775}

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